In English (and in many other languages), there is a tendency for nouns referring to liquids (water, juice), powders (sugar, sand), or substances (metal, wood) to be mass nouns, and for nouns referring to objects or people to be count nouns. This is not a hard-and-fast rule, however; such mass nouns as furniture and cutlery, which represent more easily quantified objects, show that the mass/count distinction should be thought of as a property of the terms themselves, rather than as a property of their referents. For example, the same set of chairs can be referred to as "seven chairs" and as "furniture"; though both chair and furniture are referring to the same thing, the former is a count noun and the latter a mass noun.
For another illustration of the principle that the count/non-count distinction lies not in an object but rather in the expression that refers to it, consider the English words "fruit" and "vegetables". The objects that these words describe are, objectively speaking, similar (that is, they're all edible plant parts); yet the word "fruit" is (usually) non-count, whereas "vegetables" is a plural count form. One can see that the difference is in the language, not in the reality of the objects. Meanwhile, German has a general word for "vegetables" that, like English "fruit", is (usually) non-count: das Gemüse. British English has a slang word for "vegetables" that acts the same way: "veg" [rhymes with "edge"].
An expression P has cumulative reference if and only if for any X and Y:
Consider, for example cutlery: If one collection of cutlery is combined with another, we still have "cutlery." Similarly, if water is added to water, we still have "water." But if a chair is added to another, we don't have "a chair," but rather two chairs. Thus the nouns "cutlery" and "water" have cumulative reference, while the expression "a chair" does not. (The expression "chairs", however, does.) The distinction between nouns that have cumulative reference and those that do not can be seen to correspond to the one between mass and count nouns.
An expression P has quantized reference if and only if, for any X:
This can be seen to hold in the case of the noun house: no proper part of my house, for example the bathroom, or the entrance door, is itself a house. Similarly, no proper part of a man, say his index finger, or his knee, can be described as a man. Hence, house and man have quantized reference. Collections of cutlery or cattle, however, may well have proper parts that can be described as cutlery or cattle. Hence cutlery and cattle do not have quantized reference.
Some expressions are neither quantized nor cumulative. Examples of this include collective nouns like committee. A committee may well contain a proper part which is itself a committee. Hence this expression isn't quantized. It isn't cumulative, either: the sum of two separate committees isn't necessarily a committee. In terms of the mass/count distinction, committee behaves like a count noun. Such examples indicate that the best characterization of mass nouns is that they are cumulative nouns. Count nouns should then be characterized as non-cumulative nouns: this characterization correctly groups committee together with the count nouns. If, instead, we had chosen to characterize count nouns as quantized nouns, and mass nouns as non-quantized ones, then we would (incorrectly) be led to expect committee to be a mass noun.
Many English count nouns can be used as mass nouns, and in these cases, they take on cumulative reference. For example, one may say that "there's apple in this sauce," and then apple has cumulative reference, and, hence, is used as a mass noun. Conversely, "fire" is generally a mass noun, but "a fire" refers to a discrete entity, and does not satisfy the criterion for cumulative reference. Two common situations of this process are when speaking of either servings/measurements of a substance ("Two waters please") or of several types/varieties ("waters of the world"). One may say that mass nouns that are used as count nouns are "countify" and that count ones that are used as mass nouns are "massify." Some mass nouns can't easily be countified, and some count nouns are hard to massify. For example the count noun "house" is difficult to use as mass, and the mass noun "cutlery" is hard to countify:
Another difference between mass and count nouns is the distinction between the words much and many, and between less and fewer in formal English.
There is often confusion about the two different concepts of collective noun and mass noun. Generally, collective nouns are not mass nouns, but rather are a special subset of count nouns. However, the term "collective noun" is often used to mean "mass noun" (even in some dictionaries), because users confound two different kinds of verb number invariability: (a) that seen with mass nouns such as "water" or "furniture", with which only singular verb forms are used because the constituent matter is grammatically nondiscrete (although it may ["water"] or may not ["furniture"] be etically nondiscrete); and (b) that seen with collective nouns, which is the result of the metonymical shift between the group and its (both grammatically and etically) discrete constituents.
Some words, including "mathematics" and "physics", have developed true mass-noun senses despite having grown from count-noun roots.